So, how come you’re the only one who has to go? If you have to go, so does your sister, unless your mother doesn’t count her as family. (Wait, were you living with your mom for a while before you guys got married? “And family” is usually referring to all the members of a household, so maybe that’s why she’s thinking it means just you two.) In any event, nobody is ever under an obligation to attend any function just because they were invited.
The shower in question may very well be a baby shower rather than anything wedding-related. Usually baby showers are just for the first kid (sometimes the second, too, but most people draw the line after that), but I’ve never known that to stop people who are, well, crass enough to throw a shower for themselves.
The marital status of the mother-to-be is not an issue when holding a baby shower, at least not in modern society. There was a time when premarital pregnancy was a shameful burden, but those days are pretty much over, and good riddance. (Although I admit that I’m old-fashioned enough to still look a little askance at people who have multiple “baby daddies” without ever tying the knot. It’s judgemental and probably silly, but I never claimed to be perfect.)
And no, a wedding invitation is NOT the place to be inviting people to any sort of shower. For one thing, it’s inappropriate to throw gift-giving parties for yourself. It makes you look greedy. If you’re going to get hard-core about the etiquette thing, it’s inappropriate for someone in your family to throw such a party either. For another thing, it is inappropriate to mention gifts in any way in a wedding invitation. (Yes, yes, I know some people love the registry cards in invitations, but I’m talking about traditional etiquette, not some people’s preferences.) It is also inappropriate for anyone but the host/hostess to invite people to any sort of gathering (unless, of course, you are specifically given permission to bring extra folks along).
By including shower invites with her wedding invitations, she’s either hosting her own shower, or issuing invitations to someone else’s party. Either of these are inappropriate under any circumstances, doubly so when done in conjunction with a wedding invitation.